Isosceles Trapezoid

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Isosceles Trapezoid

The isosceles trapezoid is, in fact, a trapezoid (that is, a four-sided polygon with two of them - the bases - being parallel), with two of its sides being equivalent and with its base angles of equal magnitude.

In the trapezoid, as is known, there are two bases and, each base has two base angles adjacent on both sides. In other words, in the isosceles trapezoid, there are two sets of equal base angles, as can be seen in the following illustration:

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Do isosceles trapezoids have two pairs of parallel sides?

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Definition of an isosceles trapezoid

The isosceles trapezoid is, in fact, a trapezoid (that is, a four-sided polygon with two of them - the bases - being parallel), with two of its sides being equivalent and with its base angles of equal magnitude.

In the trapezoid, as is known, there are two bases and, each base has two adjacent base angles on both sides. In other words, in the isosceles trapezoid, there are two sets of equal base angles, as can be seen in the following illustration:


Isosceles Trapezoid Properties

The properties detailed here are the unique characteristics of isosceles trapezoids among all other types of trapezoids. The following illustration describes the theorems in the best way:

  • The sides that are not parallel are congruent, that is, they have the same measure. That is, it is fulfilled: NK=LMNK=LM
  • In the isosceles trapezoid, there are two sets of equal angles for the larger base and for the smaller base.

That is, angles LL and KK are equivalent, just like angles MM and NN are also.

  • The two diagonals of the isosceles trapezoid are equal. That is, it is fulfilled: KM=LNKM=LN
  • Any isosceles trapezoid can be inscribed in a circle

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Demonstration of the Isosceles Trapezoid

To demonstrate that a trapezoid is isosceles, we must make use of the properties specified earlier, in fact, these are reciprocal theorems. It is sufficient to demonstrate just one property.

That is, if we prove that:

  • The two sides that are not parallel are congruent

or

  • The angles at the base of the trapezoid are congruent

or

  • The diagonals of the trapezoid are congruent

then, said trapezoid is an isosceles trapezoid.


The diagonals of an isosceles trapezoid

The following properties refer to the diagonals of the isosceles trapezoid. To highlight these properties in the best way, we will use this illustration:

  • The two diagonals are equal. That is, it holds that: PS=RTPS=RT
  • The triangles PTSPTS and RSTRST are congruent according to the side - side - side congruence theorem
  • The triangles TPRTPR and SRPSRP are congruent according to the side - side - side congruence theorem
  • The triangles PKRPKR and TKSTKS are isosceles triangles with all that this implies
  • The angles P1P1, R1R1, S1S1, and T1T1 are equivalent

Do you know what the answer is?

Calculation of the Area of an Isosceles Trapezoid

The calculation of the area of an isosceles trapezoid is done exactly in the same way as the area of any other trapezoid is calculated.

That is, the lengths of the two bases are added, the total sum is multiplied by the height, and then, it is divided by 2 2 .

We will use this illustration to explain the steps of the calculation:

The formula to calculate the area of the isosceles trapezoid (not exclusively) is:

A=(AB+DC)×H2A=\frac{ ( AB+ DC) \times H}{2}


Examples and exercises with isosceles trapezoids

Exercise No 1

Given the isosceles trapezoid described in the following scheme.

It is known that the sum of three angles is 240º 240º degrees.

According to the data, we must calculate all the angles of this isosceles trapezoid.

the sum of three angles is 240 degrees.

Solution:

If the sum of the three angles of the given trapezoid is 240º 240º and the total sum of the angles of a trapezoid (as with any quadrilateral) is 360º 360º , we can deduce that the fourth angle measures 120º 120º degrees.

It is one of the adjacent angles to the smaller base, let's assume angle AA. Since it is an isosceles trapezoid, the angles at the base are congruent, therefore, angle BB also measures 120º 120º .

Remember that it is a trapezoid and that the bases ABAB and DCDC are parallel, that is, angles AA and DD (just like BB and CC) are collateral angles and, therefore, complement each other and together measure 180º 180º degrees. Therefore, it will give us that angles CC and DD measure 60º 60º degrees.

Answer:

The angles of the trapezoid are 120º,120º,60º,60º120º, 120º, 60º, 60º


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Exercise No 2

Given the isosceles trapezoid described in the following scheme.

It is known that the sum of two of its sides is 120º 120º .

According to the data, we must calculate all the angles of this isosceles trapezoid.

the sum of two of its sides is 120º

Solution:

Let's go back to the rules related to the base and remember that it is a trapezoid and that the bases AB AB and DC DC are parallel, that is, the angles AA and DD (as well as B B and CC) are collateral angles and, therefore, complement each other and together measure 180º 180º degrees. Given that the amplitude we have is 120º 120º degrees, we can deduce that these are not adjacent angles on the same side (i.e., unilateral), but angles that share the same base.

Being an isosceles trapezoid, the base angles are congruent, therefore, each of them measures 60º 60º degrees.

The complementary angle (to reach 180º 180ºdegrees) of each of these angles measures 120º 120º degrees.

Answer:

The angles of the trapezoid are 120º,120º,60º,60º120º, 120º, 60º, 60º


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